Perturbation Analysis Of The Solutions For The Nonlinear Schr(?)dinger Equation

This paper mainly studies the solutions of the nonlinear Schrodinger equation with a small parameter; gives the properties of the eigenstates for the self-adjoint operator, namely, the orthogonality and completeness; introduces the perturbation theory in which people get the approximate solution of differential equations. And beginning with a perturbed NLS equation, using a multi-scales perturbation expansion, we get the zero order and the first order equations, discuss the eigenstates of the operator in the equations, induct relevant "derivative states", form the completeness of the bounded eigenstates of the associated operator in LI space, and expand the corresponding parameters in the closure, get a series evolution equations of the coefficients in the expanded formulas, find the first order approximate solution by researching the evolution equations. This paper also gives the basis of this method-the completeness we have formed and the singular perturbation technique. Since we use different time-scales in the calculation, the solution we get is more precise. It is worth pointing out that this method also can be used in other NLS equations with any perturbed terms.